How to classify
With 0 as the boundary, we divide integers into three categories: 1. A positive integer, that is, an integer greater than 0, such as 1, 2,3, until N. 2.0 is neither a positive integer nor a negative integer, but a number between positive and negative integers. 3. Negative integers, that is, integers less than 0, such as-1, -2, -3, until-n.
Edit this positive integer.
It has been a tool for human counting since ancient times. It can be said that the process of abstracting from "one cow and two cows" or "five people and six people" into positive integers is quite natural. In fact, we sometimes call positive integers natural numbers.
zero
Not only means "none", but also a symbol indicating vacancy. In ancient China, when counting chips to calculate and manipulate numbers, there was no room for counting chips. Although there is no space mark, it still creates good conditions for counting and four operations. The zero in Indo-Arabic numerology comes from the Indian character Sunya, whose original meaning is also "empty" or "blank".
negative integer
China first introduced negative numbers. The "positive and negative numbers" discussed in Chapter Nine Arithmetic Equations are the addition and subtraction of integers. The need for subtraction also promotes the introduction of negative integers. The subtraction operation can be regarded as solving the equation A-B = C. If both A and B are natural numbers, then the given equation may not have a natural number solution. In order to make it always have a solution, it is necessary to expand the natural number system into an integer system.
Edit algebraic properties of this paragraph.
The following table gives the basic properties of addition and multiplication of any integer a, b and c. Attribute addition multiplication
Closure a+b is an integer, and a × b is an integer.
The associative law a+(b+c) = (a+b)+c is an integer a × (b × c) = (a × b) × c is an integer.
The commutative law a+b = b+aa× b = b× a.
The unit element a+0 = a a × 1 = a exists.
There is an inverse a+(-a) = 0 in the integer set, and only 1 or-1 has an integer inverse for multiplication.
Distribution law a × (b+c) = a × b+ a × c
Properties and applications of editing integers in this paragraph.
Divisibility of integers
Unless otherwise specified, the concept and nature of divisibility are integers, and the letters used also represent integers. Definition: let a and b be given numbers, and b≠0. If there is an integer C that makes a=bc, it is called B divisible by A, and it is called b|a, where B is a divisor (factor) and A is a multiple of B. If there is no above C, it is said that B cannot be divisible by A. Some numerical characteristics of integers (that is, common conclusions) (1) If the last bit of an integer can be divided by 2 (or 5), (2) If the sum of the numbers of an integer can be divisible by 3 (or 9), then this number can be divisible by 3 (or 9), otherwise it can't; (3) If the last two digits of an integer can be divisible by 4 (or 25), then this number can be divisible by 4 (or 25), otherwise it can't; (4) If the last three digits of an integer can be divisible by 8 (or 125), then this number can be divisible by 8 (or 125), otherwise it can't; (5) If the difference between the sum of numbers on odd digits and the sum of numbers on even digits of an integer is a multiple of 1 1, then this number can be divisible by 1 1, otherwise it cannot.
Integer parity
(1) Odd odd = even, even even = even, odd even = odd, even × even = even, odd× even = even, odd× odd = odd; That is, the sum, difference and product of any number of even numbers are still even, the sum and difference of odd numbers are still odd, the sum and difference of even numbers are even, the sum of odd numbers and even numbers is odd, and the sum is even; (2) Even squares can be expressed in the form of (8m+ 1), and even squares can be expressed in the form of 8m or (8m+4); (3) If the product of a finite number of integers is odd, then every integer is odd; If the product of a finite number of integers is even, at least one of these integers is even; The sum and difference of two integers have the same parity; If the square root of an even number is an integer, it must be an even number.
Complete square
A complete square number and its properties can be expressed as the square of an integer, which is called a complete square number. The square number has the following properties and conclusions: (1) The single digit of the square number can only be 0, 1, 4, 5, 6, 9; (2) Even squares are multiples of 4, and odd squares are divided by 8 and 1, that is, the remainder of any square divided by 4 can only be 0 or1; (3) Ten digits with odd squares are even; (4) The single digit of a square with odd ten digits must be 6; (5) The square of a number that is not divisible by 3 is 1, and the square of a number that is divisible by 3 is divisible by 3. Therefore, if the square number is divided by 9, the remainder is 0, 1, 4, 7, and the remainder of the sum of digits of this square number divided by 9 can only be 0, 1, 4, 7; (6) The divisor of a square number is odd; (7) The product of any four consecutive integers plus 1 must be a square number. (8) Let the product of positive integers A and B be a positive integer of power of K (k≥2), and if (a, b) = 1, then both A and B are powers of K. Generally speaking, let the product of positive integers A, B and C be a positive integer of power of K (k≥2). If a, b, c ... are paired prime numbers, then a, b, c ... are all positive integers to the power of k. ..